Enhanced dissipation and Lyapunov exponents for stochastic transport-diffusion equations with small molecular diffusivity and small noise intensity

The paper deals with linear passive scalar transport-diffusion equation subject to a velocity field which is white noise in time and is mainly active at small scales in space. The purpose is investigating the enhancement of dissipation and decay given by the small-scale transport. We modify and improve estimates from the previous work by Flandoli et al. (J. Differ. Equations 394, 237–277, 2024), in order to investigate a different regime, namely small molecular diffusion and small noise intensity – corresponding to small turbulent kinetic energy. The noise specification is carefully tuned to Kolmogorov theory of turbulent fluids.